@Baldimo,
Gosh.. You are arguing that no democratic Republic existed at the time of the writing of the US Constitution and I agree. But all those things I listed now do exist. The US is a democratic Republic under today's definition. Wyoming exists under today's reality.
The times change. Meanings change. Even Hamilton used the phrase "pure democracy" when describing what the US Constitution didn't create. Within 20 year of the founding documents, people, including some of those involved in the founding documents, were using democratic to describe the US government.
The US is a democratic Republic. It is a representative democracy. It is a Republic. It is all of the above. Any claim to the contrary is using equivocation to deny it.